Discharge domains regulation and dynamic processes of direct-current triboelectric nanogenerator

Direct-current triboelectric nanogenerators arising from electrostatic breakdown can eliminate the bottleneck problem of air breakdown in conventional triboelectric nanogenerators, offering critical benefits of constant-current output, resistance to electromagnetic interference, and high output power density. Previous understanding is that its output characteristics are described by a capacitor-breakdown model or dictated by one or two discharge domains in direct-current triboelectric nanogenerators. Here, we demonstrate that the former holds only for ideal conditions and the latter cannot fully explain the dynamic process and output performance. We systematically image, define, and regulate three discharge domains in direct-current triboelectric nanogenerators, then a “cask model” is developed to bridge the cascaded-capacitor-breakdown dynamic model in ideal conditions and real outputs. Under its guidance, the output power is increased by an order of magnitude within a wide range of resistive loads. These unexplored discharge domains and optimization methods revolutionize the output performance and potential applications of direct-current triboelectric nanogenerators.

Statistical result of the charge transferred between FE and CCE, which was normalized with the sliding area. The error bar represents the standard deviation (six trials). Source data are provided as a Source data file.  Compared with a hydroelectric energy generating system, FE, the external circuit, and CCE can be seen as a triboelectric power generating system. Based on the contact electrification effect, charges go into the triboelectric power generating system and leave the system due to the electrostatic breakdown phenomenon. The working path of those charges resembles that of a working fluid in a hydroelectric energy generating system (Fig. 1a). The work done on a hydroelectric system fills the container with fluid and inputs potential energy to the whole system, which is manifested in the DC-TENG system as friction between FE and the TL does work and generates charges into the system ( Fig. 1b(i)). Under the gravity field, the water flows out and outputs power, for example, by driving the water wheel as demonstrated in Fig. 1a. Likewise, when the charged TL moves below CCE, the opposite charges are induced in CCE, resulting in a strong electric field between CCE and TL. Once the electric field intensity is higher than the breakdown threshold, the triboelectric charges on CCE and TL will discharge in the air 1 , doing work in the external circuit ( Fig. 1b(ii)). When the fluid level moves down below the outlet due to the decrease or stop of the inlet, the fluid will no longer do work to the outside. The identical condition occurs in the DC-TENG system, i.e., no current output after the relative motion stops ( Fig. 1b(iii)). Fig. 1b demonstrates a working cycle, and the wire with a load in this diagram indicates the external circuit.

Supplementary Note 2 Note on the surface charge
As some previous work declared, the bipolar, multipolar, and mosaics of charges on tribo surfaces occur in micro and macro scales, especially for the contact between two dielectric layers. 2, 3 However, it is worth noting that these effects can be simplified to an equivalent charge, specifically the net charge, when applying contact electrification to TENG. This work focuses more on the output electrical performance of TENG, and the measured output signal is the result of the electric field generated by the superposition of all charges present in the entire area. In practice, regardless of how the local bipolar, multipolar, and mosaic patterns perform, the overall equivalent charge remains relatively stable. Hence, under this condition, for the sake of simplicity, the terms such as "charge", "surface charge density", "residual charge", and other similar expressions associated with electrostatic charges generated by contact electrification in this work refer to a simplified expression of "net charge", "net surface charge density", "net residual charge", and so forth.

Supplementary Note 3 Determination of the threshold of the electric field intensity base on Paschen's law
Paschen's law for gas breakdown gives an empirical relation between the breakdown voltage and the product of the gas pressure and the distance between the two electrodes involved in the breakdown process. The question is shown in Equation S1.
where Vb is the breakdown voltage; p is the gas pressure; d is the distance; A and B are constants determined by the pressure and composition of the gas; and γ is the secondary electron emission coefficient.
In our manuscript, the working condition was in the air of 1 atmosphere, so we plotted While researchers generally accept the validity of the Paschen's law for distances above 5-10 microns (i.e., the right side of the curve in Supplementary Fig. 1), there is ongoing debate regarding its applicability for smaller air gaps (i.e., the left side of the curve in Supplementary Fig. 1) [4][5][6] , recently. In our work, we have to utilize the breakdown voltage at distances ranging from ~ 0 (the contact point of the friction electrode (FE) and the triboelectric layer (TL)) to millimeter scale (the distance between FE and charge collecting electrode (CCE)), which falls within the controversial region where the accuracy of Paschen's law is debated. We derived the breakdown threshold of the electric field intensity from the calculated breakdown voltage, which varied from 3 MV m -1 to 7 MV m -1 for the right side of Paschen's curve. The DC-TENG device itself presents many non-ideal factors that further complicate the analysis of its electrical properties. For instance, the electric field around the DC-TENG is not uniform as assumed in Paschen's law, which is probably due to the mechanical configuration, materials selections, and electromagnetic environment issues. To simplify our model and emphasize the main focus of this work, we used a breakdown voltage magnitude that many researchers have employed, which is 3 MV m -1 5, 7-9 .
For the left side of the curve, some works stated that in addition to the Townsend avalanche of gaseous ions, other effects such as field emission of electrons and tunneling electrons have been proposed as potential factors, and they offered a modified Paschen's curve with a slope of 60-80 MV m -1 4-6 . Whether considering the original Paschen's curve or the modified one, the breakdown electric field intensity exceeds 3 MV m -1 by more than an order of magnitude, which does not align with real-world conditions. For instance, the electric field intensity threshold for triggering the field emission is ~75 MV m -1 5 , requiring ultra-high surface charge density of ~1330 μC m -2 , which is a strict condition to achieve in the air with 1 atmosphere. Hence, 3 MV m -1 can be a proper magnitude of breakdown threshold.

Supplementary Note 4 Calculation for a single capacitor-breakdown model
Before the electrostatic breakdown occurs, the focusing region at CCE and TL can be treated as a parallel capacitor. The equation for calculating the capacitance is shown as following: where denotes the permittivity of the material, which is air in this application, between CCE and TL, denotes facing area of a parallel capacitor, denotes the vertical distance between the two parallel plates, CCE denotes the length of CCE in the direction along the sliding direction, CCE denotes the width of CCE perpendicular the sliding direction, and CCE−TL denotes the vertical distance between CCE and TL.
Further, the voltage between CCE and TL CCE−TL can be calculated with the charge in CCE and TL CCE−TL : Calculated from the voltage, the electric field intensity between CCE and TL is When CCE−TL is larger than the breakdown threshold in the air, which is 3 MV m -1 , the focusing region at CCE and TL can be seen as a resistor CCE−TL . Then the output current can be obtained as Taking the sliding speed as an example variable and keeping other variables fixed, calculated by the single capacitor-breakdown model, low sliding speed leads to pulse signals ( Supplementary Fig. 2a). As the input increases, the frequency of the pulse increases ( Supplementary Fig. 2b and 2c), and when the input increases to a specific value, a constant current output is obtained (Supplementary Fig. 2d).

Supplementary Note 5 Analysis of the three main variables
Charges inlet on CCE CCE can be formulated as a product of the surface charge density on TL CCE and the swept area CCE−TL .
Since the production of the charge is continuous as FE keeps sliding on TL, the rate of charge inlet is the derivative of equation S6.
where CCE−TL is the relative displacement of CCE and TL, and the derivative of it is thus the sliding speed sliding . Hence, the inlet rate of charges on CCE−TL can be modeled by the following equation.

Supplementary Note 6 The random factor in the capacitor breakdown model
The roughness of TL can affect contact electrification performance by altering the real area of contact and contact pressure. Considering the charge generation side, a random factor is employed to take practically inconsistent conditions into account, such as the uniformity of materials, the roughness of two contacting surfaces, and contact efficiency. It conforms to the normal distribution (Supplementary Fig. 3a). With this random factor, the rate of charge flowing into the capacitor is not a constant ( Supplementary Fig. 3b), but a value with kind of randomness (Supplementary Fig.   3c), which is consistent with the practical conditions.  Fig. 4a). To be more specific, by dividing the whole focusing region between CCE and TL into two categories due to the statistical analysis in the microscopic view, the area with the distance of min to

Supplementary Note 7 The division strategy for secondary capacitors in the cascaded-capacitor-breakdown model
is divided into one category, and that of ( max − min ) 2 to max is divided into another category (Supplementary Fig. 4b). Similarly, Supplementary Fig. 4c and Supplementary Fig. 4d illustrate the situation when more categories are classified.

Supplementary Note 8 Derivation for the breakdown threshold in charge of a single capacitor unit
Assuming each capacitor unit is a parallel capacitor, combining equations S2, S3, and S4, the electric field intensity between each capacitor unit unit is derived as where unit is the charge on both sides of the parallel capacitor unit and unit is the facing area of a single parallel capacitor unit, determined by the number of the parts the whole capacitor is divided into.
From equation S9, electric field intensity is irrelated to the vertical distance between the two parallel plates. Hence all the capacitor units have the same breakdown threshold.
Further, the judgment threshold in charge shown in Fig. 1d can be calculated as threshold = unit threshold (S10) The breakdown threshold for capacitor units with different vertical separation distances is the same. After the breakdown threshold is reached, the output current for a single capacitor unit can be calculated as unit = unit unit unit (S11) Since the secondary capacitors are divided by the real-world distance between CCE and TL, the after-breakdown resistance for each secondary capacitor varies.

Supplementary Note 9 Determination of parameter n
As some parameters in the cascaded-capacitor-breakdown simulation model require pre-experimental calibration, experimental data at two velocities (1 mm s -1 and 75 mm s -1 ) was used as the tuning set, while experimental data at four velocities (5, 10, 25, and 50 mm s -1 ) is used as the test set. CCE was determined by pre-experiments ( Supplementary Fig. 6).  Fig. 7a and 7b, where the subscript "E" represents experimental data, the subscript "S" represents the simulated results, and the subscript "k" indicates the dimension of the tuning set). The general trend is the difference between the experimental data and the simulated results decreases with an increasing (Supplementary Fig. 7c and 7d). Nevertheless, cannot be infinitely large, since the simulation time is proportional to the value of and the computing resources are finite ( Supplementary Fig. 8). Comprehensively considering the statement above, = 35 was chosen for this experiment. For future applications, with more data, the determination of parameter can be modified by increasing the tuning set size (the value of ) and the range of , as shown in Supplementary Fig. 7a and Supplementary Fig. 7b.

Supplementary Note 10 Calculation for root-mean-square (RMS) current and crest factor (CF)
The root-mean-square (RMS) current mentioned in this work is defined as the value of the constant current that dissipates the same power in a resistor as the time-varying current does by our DC-TENG, which can also be called effective current. The equal derived from the energy equation is shown as follows: where ( ) is a set of current data measured within time .
The crest factor (CF) mentioned in this work is defined as the ratio of peak current to the effective current, indicating how extreme the peak value is in a current waveform.
A CF of 1 suggests no peaks in the current signals, indicating a constant-current output.

The equation used in this work is
where max is the maximum value in a set of current data.

Supplementary Note 11 Note on the examination of the variable
Variable CCE is obtained by measuring the charges collected at CCE, and dividing this value with the area swept by the relative motion. It is mainly influenced by charging and discharging processes of CCE−TL . In the charging process, part of the charges generated by the friction between FE and TL flows into this equivalent capacitor. In the discharging process, these charges participate the electrostatic breakdown at CCE.
Hence, CCE is constrained by the contact electrification and electrostatic breakdown properties of the material-pairs 10 . In this work, both contact electrification and electrostatic breakdown are achieved with a copper electrode and a dielectric layer.
Therefore, we kept the electrode unchanged and altered the material of the dielectric layer to regulate CCE .

Supplementary Note 12 Note on friction materials selection
It is worth noting that when selecting friction material-pairs, in addition to considering the surface charge density that the material-pair can generate, other properties of the material 10 , such as the friction coefficient, manufacturing process, and the sensitivity to temperature and humidity should also be considered. In the experiment, it was found that the surface charge density generated by the friction between ethylene tetrafluoroethylene (ETFE) and the copper electrode was high, but the large friction coefficient caused it difficult for FE to smoothly slide on TL. This caused the low contact efficiency and an unstable contact electrification process at low speed, resulting in a fluctuating output current. Besides, it puts unpredictable stress on the device, leading to wear and tear and shortening the life of devices.

Supplementary Note 13 Sampling point selection principle
When discussing the electric field intensity distribution in the 1 st and 2 nd domains, two sampling points were used to represent their situation. The electric field intensity is generally strongest at sharp corners, making them the most likely location for discharge to occur. However, in finite element method simulations, the calculation of the sudden change in the geometry is highly dependent on the mesh distribution, and also turns out unreliable due to geometric discontinuity. Focusing on the region we were interested in, as shown in Supplementary Fig. 9a, the mesh distribution was displayed in Supplementary Fig. 9b. The sampling point in the 2 nd domain was taken as an example to declare the reason for selecting such sampling points in finite element method simulations in the manuscript. By changing the element number of the edge of FE, which was highlighted in Supplementary Fig. 9b, the electric field intensity on each gridding point was simulated and shown in Supplementary Fig. 9c. The electric field intensities at the two ends are larger than that of the gridding points in the middle and the fluctuation with the number of elements are also greatest on these two positions (inset in Supplementary Fig. 9c). To ensure consistent simulation results and minimize the impact of the number of elements on the outcome, we selected points that were not at the two ends but close to where the maximum electric field intensity occurs as sampling points. Although the electric field intensities at these points is relatively low (Supplementary Fig. 9d), it has no influence on our determination of whether breakdown will occur in the 1 st and 2 nd domains. When the electric field intensity at a sampling point exceeds the breakdown threshold, there must be a point with an even higher electric field intensity in the corresponding discharge domain, indicating the breakdown will definitely occur. Selecting the sampling point a short distance away from a geometrically sharp point allows for more stable and reliable numerical simulation (Supplementary Fig. 9d). The discarded magnitude of this part of electric field intensity further supports the existence of the two breakdown domains we defined.

Supplementary Note 14 Analysis of the evidence of electrostatic breakdown in electrical experimental data
In a short circuit, while measuring the current data in Fig. 2, the total amount of triboelectric charge and the surface residual charge after the electrostatic breakdown were measured at various speeds using the method described in Supplementary Fig.   11, 12 and Supplementary Note 15, 16. As shown in Fig. 4k, comparing the total triboelectric charge with the charge collected in the external circuit, it is found that a large amount of charge did not flow through the external circuit. This part of the charge is prone to release from 2 nd domain through an electrostatic breakdown in the air.
In an open circuit, with relative motions, the triboelectric charges concentrated on FE lead to a significant potential difference between FE and CCE, equivalent to applying a bias voltage between the two 11 . On the one hand, this potential difference will make it difficult for triboelectric charges to enter or remain in FE. As shown in Fig. 4l, when charging a test capacitor with DC-TENG as shown in the circuit inset, as the number of working cycle increases, the amount of charge that can be charged into the test capacitor in a single cycle decrease. On the other hand, considering FE and CCE as a capacitor, the voltage between the two electrodes is proportional to the amount of charge it contains. When the charge accumulated to a certain amount, the breakdown voltage was reached, resulting in the electrostatic breakdown in the 3 rd domain and charges being released, as marked with the pentagram in Fig. 4k.

Supplementary Note 15 Measurement of the total triboelectric charges generated by the friction between FE and TL
As shown in Supplementary Fig. 11(i-iii), with relative sliding between FE and TL, the triboelectric charge is generated due to contact electrification and discharged due to electrostatic breakdown. When the DC-TENG device moves onto the back electrode, the discharge is electrostatically shielded since there is charge induced on the back electrode. By measuring the amount of charge transferred from the ground to the back electrode, the equal charge generated on the TL via contact electrification can be thus measured. From another perspective, the back electrode, dielectric layer, and the sliding FE can be seen as a single electrode sliding mode alternating current TENG (AC-TENG) 12 , and the amount of charge on the dielectric layer can be determined by measuring the transferred charge in the external circuit.

Supplementary Note 16 Measurement of the residual charges on the surface of TL after DC-TENG operation
After DC-TENG stops, an electrode connected to the ground was put onto the surface of TL. Due to electrostatic induction, positive charges will flow from the ground to the contact surface of the electrode to balance the potential difference. By measuring the amount of these induced charges, the residual surface charge can be measured. The residual surface charge density was obtained by dividing the residual surface charge with the area of the testing electrode, which is 1 cm 2 for the experiments with different widths of CCE, and 4 cm 2 for other experiments in this paper.

Supplementary Note 17 Influence of the isolation layer on contact electrification process and output signals
Although there is contact electrification effect between the Kapton tape and the PVC film, it has little influence on the output performance of the DC-TENG. Kapton is a kind of dielectric material, and after a brief rubbing, the charge on the Kapton tape becomes saturated. Since the conductive electrode on the other side of the Kapton tape serves as a back electrode to bound charges 13 , the discharge between Kapton and PVC rarely occurs. Hence, the saturated Kapton no longer participates in the contact electrification process, simply acting as an isolating tool with ignorable influence on the electric field or electric output. As shown in Supplementary Fig. 13, compared with the charge density measured in the external circuit of DC-TENG with uncovered FE (dozens of micro coulombs per square meter), that of the DC-TENG with covered FE configuration (inset of Supplementary Fig. 13) can be negligible (~1.3 μC m -2 ).

Supplementary Note 18 Note on the equivalent circuit of DC-TENG
To better understand it in a circuit, an equivalent circuit is shown in Fig. 6b. The electric properties in the 1 st , 2 nd , and 3 rd domain are similar, so FE and CCE, FE and TL can also be analyzed as capacitor models similar to the one in Fig. 1c. Based on relative position of these mechanical structure, the equivalent circuit was constructed. The current source represents the continuous inlet charges of the triboelectric system due to contact electrification. Together with generation, some charges are released through the 2 nd BD. Equivalent secondary circuits of the 1 st BD and 2 nd BD are connected in parallel while the secondary circuit of the 3 rd BD is in series to the 1 st BD. As the circuit presented, once the 3 rd breakdown releases all the charges in the system, there is no electrical output in the external circuit, though it rarely happens.

DC-TENG with end-isolated FE
It is worth noting that as the resistance in the external circuit increases, it is easier to observe the 3 rd BD for the DC-TENG with end-isolated FE compared to the conventional configuration. Visible evidence is taken with the same setting to Fig. 4h, as shown in Supplementary Fig. 10a. The bright discharge path connecting the two edges of the Kapton tape displays in Supplementary Fig. 10b proves that, on the one hand, the tape effectively protected FE from being exposed to the air, and on the other hand, the amount of charge accumulated in FE was so large that the breakdown in the 3 rd domain was triggered. Supplementary Fig. 10c provides the output current under a resistance of 6 GΩ, obviously showing the 3 rd BD. A possible explanation is that, with the 2 nd domain blocked, some charges that originally flowed out from the 2 nd domain leaves the triboelectric system from 1 st domain, and the rest stays in the system and accumulates until the breakdown threshold of the 3 rd BD is reached. Moreover, some tiny spikes on the current signals might be caused by the discharge between TL and the small strip of tape without metal electrodes on the back. Further modifications to the fabrication are required to solve this issue.

Supplementary Note 20 Influence of environmental conditions
For the sake of controlling variables and simplifying the theoretical model and experiments to a certain extent, consistent environmental parameters were employed throughout this work. However, it is worth noting that in practical applications, both temperature and relative humidity can impact the operation of DC-TENG. The presence of adsorbed water on the tribo surfaces is crucial for both the charging and discharging processes 14 . Its role in the triboelectrification process depends on the motion mode (contact-separation mode, sliding or free-standing mode), materials (chemical and physical properties characterized by the contact angle, functional group, roughness, and inherent triboelectrification ability), and other operational parameters. For discharging process, humidity may alter the breakdown characteristics of the gas, resulting in a noticeable decrease in the breakdown voltage as humidity increases 15 . As the DC-TENG device arises from both triboelectrification and electrostatic discharge, its output performance is influenced by both physical processes. Some research suggested that the output performance of DC-TENG is improved in high-humidity environments, as a high-humidity environment not only promotes the electrification effect of the hydrophobic materials to generate more triboelectric charges but also reduces the gap breakdown voltage, facilitating the electrostatic breakdown process. Nevertheless, the complex influence of humidity on the two physical processes is still controversial, and its influence on the output performance of DC-TENG needs more comprehensive study.

Supplementary Note 21 Generalization for the definition of discharge domains
To better explain and further utilize electrostatic breakdowns occurring in this triboelectric power generation, we generalized and presented more explanations about the concept of discharge domain for a conventional DC-TENG. In addition to the domains defined above, electrostatic breakdown may also occur at other locations due to manufacturing process, materials selection, and geometry design, especially where there is a sharp corner geometry (Supplementary Fig. 16).
These locations can also be classified into the categories defined above. The discharge participated by the conductor between the bottom of CCE and the electrical appliance in the external circuit can be classified into the breakdown in the 1 st domain since it has a positive impact on the output of a conventional DC-TENG. It is because whichever the discharge on CCE or the wire, the negative charge tends to flow to FE to balance the potential difference continuously generated by contact electrification, which causes the charge to flow through the external circuit and does work externally. The discharge participated by the conductor between the contact electrification interface of FE and the electrical appliance in the external circuit can be classified into the breakdown in the 2 nd domain since it has a negative effect. The reason is both the generation of the whole triboelectric charges and the release of charges in the 2 nd domain happens before the electrical appliance, which means the amount of charge that flows through the electrical appliance is the net charge that equals the total triboelectric charge minus the charge released by 2 nd breakdown, much less than the original triboelectric charge, restricting the output performance.